Page 96 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH02_95_08/30/05

Basic Members: Lumped- and Distributed-Parameter Modeling and Design

Basic Members: Lumped- and Distributed-Parameter Modeling and Design 95
2
2
ds = dx + du 2 z (2.175)
The increase in length due to the action of the axial force can be ex-
pressed as

2
du
du
ds + dx ( dx ) 2 (2.176)
1
z
z
ds í dx = 싉 2 dx
The work performed by the axial force is converted to strain energy and
is equal to
F x l du z 2
U = 2 ฒ ( dx ) dx (2.177)
a
0
and therefore the total strain energy (the sum of bending and axial
strain energy contributions) is
2
EI y l d u z 2 F x l du z 2
U = U + U = 2 ฒ ( ) dx + 2 ฒ ( dx ) dx (2.178)
a
b
2
0 dx 0
Assuming the deflection varies according to the sinusoidal law u z
sin (Ȧt), the kinetic energy of the vibrating microcantilever is
2
Ȧ ȡA l 2
T = 2 ฒ u dx (2.179)
z
0
Two approximate methods are presented next which are designed to
determine the bending-related resonant frequency of a microcantilever
under the action of a tip axial load.
5
Rayleigh’s procedure. According to Rayleigh (see Timoshenko, for in-
stance) the natural frequency can be found by enforcing U = T and
by considering a certain shape of the deformed cantilever. When the
deflection curve is the one produced by a force F 1z acting at the free tip,
namely,

F 1z l 3 3x x 3
u (x) = 1 ෹ + (2.180)
z 3EI y 2l (2l) 3
the altered resonant frequency becomes

5EI )
( 2F l 2 EI y
x
Ȧ = 3.56753 1+ y ml 3 (2.181)

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